where were bounded multiplicative functions, were fixed shifts, was a quantity going off to infinity, and was a generalised limit functional. Our main technical result asserted that these correlations were necessarily the uniform limit of periodic functions . Furthermore, if (weakly) pretended to be a Dirichlet character , then the could be chosen to be –isotypic in the sense that whenever are integers with coprime to the periods of and ; otherwise, if did not weakly pretend to be any Dirichlet character , then vanished completely. This was then used to verify several cases of the logarithmically averaged Elliott and Chowla conjectures.

The purpose of this paper was to investigate the extent to which the methods could be extended to non-logarithmically averaged settings. For our main technical result, we now considered the unweighted averages

where is an additional parameter. Our main result was now as follows. If did not weakly pretend to be a twisted Dirichlet character , then converged to zero on (doubly logarithmic) average as . If instead did pretend to be such a twisted Dirichlet character, then converged on (doubly logarithmic) average to a limit of -isotypic functions . Thus, roughly speaking, one has the approximation

for most .

Informally, this says that at almost all scales (where “almost all” means “outside of a set of logarithmic density zero”), the non-logarithmic averages behave much like their logarithmic counterparts except for a possible additional twisting by an Archimedean character (which interacts with the Archimedean parameter in much the same way that the Dirichlet character interacts with the non-Archimedean parameter ). One consequence of this is that most of the recent results on the logarithmically averaged Chowla and Elliott conjectures can now be extended to their non-logarithmically averaged counterparts, so long as one excludes a set of exceptional scales of logarithmic density zero. For instance, the Chowla conjecture

is now established for either odd or equal to , so long as one excludes an exceptional set of scales.

In the logarithmically averaged setup, the main idea was to combine two very different pieces of information on . The first, coming from recent results in ergodic theory, was to show that was well approximated in some sense by a nilsequence. The second was to use the “entropy decrement argument” to obtain an approximate isotopy property of the form

for “most” primes and integers . Combining the two facts, one eventually finds that only the almost periodic components of the nilsequence are relevant.

In the current situation, each is approximated by a nilsequence, but the nilsequence can vary with (although there is some useful “Lipschitz continuity” of this nilsequence with respect to the parameter). Meanwhile, the entropy decrement argument gives an approximation basically of the form

for “most” . The arguments then proceed largely as in the logarithmically averaged case. A key lemma to handle the dependence on the new parameter is the following cohomological statement: if one has a map that was a quasimorphism in the sense that for all and some small , then there exists a real number such that for all small . This is achieved by applying a standard “cocycle averaging argument” to the cocycle .

It would of course be desirable to not have the set of exceptional scales. We only know of one (implausible) scenario in which we can do this, namely when one has far fewer (in particular, subexponentially many) sign patterns for (say) the Liouville function than predicted by the Chowla conjecture. In this scenario (roughly analogous to the “Siegel zero” scenario in multiplicative number theory), the entropy of the Liouville sign patterns is so small that the entropy decrement argument becomes powerful enough to control all scales rather than almost all scales. On the other hand, this scenario seems to be self-defeating, in that it allows one to establish a large number of cases of the Chowla conjecture, and the full Chowla conjecture is inconsistent with having unusually few sign patterns. Still it hints that future work in this direction may need to split into “low entropy” and “high entropy” cases, in analogy to how many arguments in multiplicative number theory have to split into the “Siegel zero” and “no Siegel zero” cases.

One would certainly like to extend these results to higher order correlations than the two-point correlations. This looks to be difficult (though perhaps not completely impossible if one allows for logarithmic averaging): in a previous paper I showed that achieving this in the context of the Liouville function would be equivalent to resolving the logarithmically averaged Sarnak conjecture, as well as establishing logarithmically averaged local Gowers uniformity of the Liouville function. However, in this paper we are able to avoid having to resolve these difficult conjectures to obtain partial results towards the (logarithmically averaged) Chowla and Elliott conjecture. For the Chowla conjecture, we can obtain all odd order correlations, in that

for all odd and all integers (which, in the odd order case, are no longer required to be distinct). (Superficially, this looks like we have resolved “half of the cases” of the logarithmically averaged Chowla conjecture; but it seems the odd order correlations are significantly easier than the even order ones. For instance, because of the Katai-Bourgain-Sarnak-Ziegler criterion, one can basically deduce the odd order cases of (2) from the even order cases (after allowing for some dilations in the argument ).

For the more general Elliott conjecture, we can show that

for any , any integers and any bounded multiplicative functions , unless the product weakly pretends to be a Dirichlet character in the sense that

This can be seen to imply (2) as a special case. Even when does pretend to be a Dirichlet character , we can still say something: if the limits

exist for each (which can be guaranteed if we pass to a suitable subsequence), then is the uniform limit of periodic functions , each of which is –isotypic in the sense that whenever are integers with coprime to the periods of and . This does not pin down the value of any single correlation , but does put significant constraints on how these correlations may vary with .

Among other things, this allows us to show that all possible length four sign patterns of the Liouville function occur with positive density, and all possible length four sign patterns occur with the conjectured logarithmic density. (In a previous paper with Matomaki and Radziwill, we obtained comparable results for length three patterns of Liouville and length two patterns of Möbius.)

To describe the argument, let us focus for simplicity on the case of the Liouville correlations

assuming for sake of discussion that all limits exist. (In the paper, we instead use the device of generalised limits, as discussed in this previous post.) The idea is to combine together two rather different ways to control this function . The first proceeds by the entropy decrement method mentioned earlier, which roughly speaking works as follows. Firstly, we pick a prime and observe that for any , which allows us to rewrite (3) as

Making the change of variables , we obtain

The difference between and is negligible in the limit (here is where we crucially rely on the log-averaging), hence

The entropy decrement argument can be used to show that the latter limit is small for most (roughly speaking, this is because the factors behave like independent random variables as varies, so that concentration of measure results such as Hoeffding’s inequality can apply, after using entropy inequalities to decouple somewhat these random variables from the factors). We thus obtain the approximate isotopy property

for most and .

On the other hand, by the Furstenberg correspondence principle (as discussed in these previous posts), it is possible to express as a multiple correlation

for some probability space equipped with a measure-preserving invertible map . Using results of Bergelson-Host-Kra, Leibman, and Le, this allows us to obtain a decomposition of the form

where is a nilsequence, and goes to zero in density (even along the primes, or constant multiples of the primes). The original work of Bergelson-Host-Kra required ergodicity on , which is very definitely a hypothesis that is not available here; however, the later work of Leibman removed this hypothesis, and the work of Le refined the control on so that one still has good control when restricting to primes, or constant multiples of primes.

Using the equidistribution theory of nilsequences (as developed in this previous paper of Ben Green and myself), one can break up further into a periodic piece and an “irrational” or “minor arc” piece . The contribution of the minor arc piece can be shown to mostly cancel itself out after dilating by primes and averaging, thanks to Vinogradov-type bilinear sum estimates (transferred to the primes). So we end up with

which already shows (heuristically, at least) the claim that can be approximated by periodic functions which are isotopic in the sense that

But if is odd, one can use Dirichlet’s theorem on primes in arithmetic progressions to restrict to primes that are modulo the period of , and conclude now that vanishes identically, which (heuristically, at least) gives (2).

The same sort of argument works to give the more general bounds on correlations of bounded multiplicative functions. But for the specific task of proving (2), we initially used a slightly different argument that avoids using the ergodic theory machinery of Bergelson-Host-Kra, Leibman, and Le, but replaces it instead with the Gowers uniformity norm theory used to count linear equations in primes. Basically, by averaging (4) in using the “-trick”, as well as known facts about the Gowers uniformity of the von Mangoldt function, one can obtain an approximation of the form

where ranges over a large range of integers coprime to some primorial . On the other hand, by iterating (4) we have

for most semiprimes , and by again averaging over semiprimes one can obtain an approximation of the form

For odd, one can combine the two approximations to conclude that . (This argument is not given in the current paper, but we plan to detail it in a subsequent one.)

as for any distinct integers , where is the Liouville function. (The usual formulation of the conjecture also allows one to consider more general linear forms than the shifts , but for sake of discussion let us focus on the shift case.) This conjecture remains open for , though there are now some partial results when one averages either in or in the , as discussed in this recent post.

A natural generalisation of the Chowla conjecture is the Elliott conjecture. Its original formulation was basically as follows: one had

whenever were bounded completely multiplicative functions and were distinct integers, and one of the was “non-pretentious” in the sense that

for all Dirichlet characters and real numbers . It is easy to see that some condition like (2) is necessary; for instance if and has period then can be verified to be bounded away from zero as .

as for any Dirichlet character . To support this conjecture, we proved an averaged and non-asymptotic version of this conjecture which roughly speaking showed a bound of the form

whenever was an arbitrarily slowly growing function of , was sufficiently large (depending on and the rate at which grows), and one of the obeyed the condition

for some that was sufficiently large depending on , and all Dirichlet characters of period at most . As further support of this conjecture, I recently established the bound

under the same hypotheses, where is an arbitrarily slowly growing function of .

In view of these results, it is tempting to conjecture that the condition (4) for one of the should be sufficient to obtain the bound

when is large enough depending on . This may well be the case for . However, the purpose of this blog post is to record a simple counterexample for . Let’s take for simplicity. Let be a quantity much larger than but much smaller than (e.g. ), and set

For , Taylor expansion gives

and

and hence

and hence

On the other hand one can easily verify that all of the obey (4) (the restriction there prevents from getting anywhere close to ). So it seems the correct non-asymptotic version of the Elliott conjecture is the following:

Conjecture 1 (Non-asymptotic Elliott conjecture) Let be a natural number, and let be integers. Let , let be sufficiently large depending on , and let be sufficiently large depending on . Let be bounded multiplicative functions such that for some , one has

for all Dirichlet characters of conductor at most . Then

The case of this conjecture follows from the work of Halasz; in my recent paper a logarithmically averaged version of the case of this conjecture is established. The requirement to take to be as large as does not emerge in the averaged Elliott conjecture in my previous paper with Matomaki and Radziwill; it thus seems that this averaging has concealed some of the subtler features of the Elliott conjecture. (However, this subtlety does not seem to affect the asymptotic version of the conjecture formulated in that paper, in which the hypothesis is of the form (3), and the conclusion is of the form (1).)

was small (for a slowly growing function of ) if was bounded and completely multiplicative, and one had a condition of the form

for some large . However, to obtain an analogous bound for (5) it now appears that one needs to strengthen the above condition to

in order to address the counterexample in which for some between and . This seems to suggest that proving (5) (which is closely related to the case of the Chowla conjecture) could in fact be rather difficult; the estimation of (6) relied primarily of prior work of Matomaki and Radziwill which used the hypothesis (7), but as this hypothesis is not sufficient to conclude (5), some additional input must also be used.

as for any fixed natural number . This conjecture remains open, though there are a number of partial results (e.g. these twoprevious results of Matomaki, Radziwill, and myself).

A natural generalisation of Chowla’s conjecture was proposed by Elliott. For simplicity we will only consider Elliott’s conjecture for the pair correlations

For such correlations, the conjecture was that one had

as for any natural number , as long as was a completely multiplicative function with magnitude bounded by , and such that

for any Dirichlet character and any real number . In the language of “pretentious number theory”, as developed by Granville and Soundararajan, the hypothesis (2) asserts that the completely multiplicative function does not “pretend” to be like the completely multiplicative function for any character and real number . A condition of this form is necessary; for instance, if is precisely equal to and has period , then is equal to as and (1) clearly fails. The prime number theorem in arithmetic progressions implies that the Liouville function obeys (2), and so the Elliott conjecture contains the Chowla conjecture as a special case.

As it turns out, Elliott’s conjecture is false as stated, with the counterexample having the property that “pretends” locally to be the function for in various intervals , where and go to infinity in a certain prescribed sense. See this paper of Matomaki, Radziwill, and myself for details. However, we view this as a technicality, and continue to believe that certain “repaired” versions of Elliott’s conjecture still hold. For instance, our counterexample does not apply when is restricted to be real-valued rather than complex, and we believe that Elliott’s conjecture is valid in this setting. Returning to the complex-valued case, we still expect the asymptotic (1) provided that the condition (2) is replaced by the stronger condition

as for all fixed Dirichlet characters . In our paper we supported this claim by establishing a certain “averaged” version of this conjecture; see that paper for further details. (See also this recent paper of Frantzikinakis and Host which establishes a different averaged version of this conjecture.)

One can make a stronger “non-asymptotic” version of this corrected Elliott conjecture, in which the parameter does not go to infinity, or equivalently that the function is permitted to depend on :

Conjecture 1 (Non-asymptotic Elliott conjecture) Let , let be sufficiently large depending on , and let be sufficiently large depending on . Suppose that is a completely multiplicative function with magnitude bounded by , such that

for all Dirichlet characters of period at most . Then one has

for all natural numbers .

The -dependent factor in the constraint is necessary, as can be seen by considering the completely multiplicative function (for instance). Again, the results in my previous paper with Matomaki and Radziwill can be viewed as establishing an averaged version of this conjecture.

Meanwhile, we have the following conjecture that is the focus of the Polymath5 project:

It is instructive to compute some near-counterexamples to Conjecture 2 that illustrate the difficulty of the Erdös discrepancy problem. The first near-counterexample is that of a non-principal Dirichlet character that takes values in rather than . For this function, one has from the complete multiplicativity of that

If denotes the period of , then has mean zero on every interval of length , and thus

Thus has bounded discrepancy.

Of course, this is not a true counterexample to Conjecture 2 because can take the value . Let us now consider the following variant example, which is the simplest member of a family of examples studied by Borwein, Choi, and Coons. Let be the non-principal Dirichlet character of period (thus equals when , when , and when ), and define the completely multiplicative function by setting when and . This is about the simplest modification one can make to the previous near-counterexample to eliminate the zeroes. Now consider the sum

with for some large . Writing with coprime to and at most , we can write this sum as

Now observe that . The function has mean zero on every interval of length three, and is equal to mod , and thus

for every , and thus

Thus also has unbounded discrepancy, but only barely so (it grows logarithmically in ). These examples suggest that the main “enemy” to proving Conjecture 2 comes from completely multiplicative functions that somehow “pretend” to be like a Dirichlet character but do not vanish at the zeroes of that character. (Indeed, the special case of Conjecture 2 when is completely multiplicative is already open, appears to be an important subcase.)

All of these conjectures remain open. However, I would like to record in this blog post the following striking connection, illustrating the power of the Elliott conjecture (particularly in its nonasymptotic formulation):

The argument relies heavily on two observations that were previously made in connection with the Polymath5 project. The first is a Fourier-analytic reduction that replaces the Erdos Discrepancy Problem with an averaged version for completely multiplicative functions . An application of Cauchy-Schwarz then shows that any counterexample to that version will violate the conclusion of Conjecture 1, so if one assumes that conjecture then must pretend to be like a function of the form . One then uses (a generalisation) of a second argument from Polymath5 to rule out this case, basically by reducing matters to a more complicated version of the Borwein-Choi-Coons analysis. Details are provided below the fold.

There is some hope that the Chowla and Elliott conjectures can be attacked, as the parity barrier which is so impervious to attack for the twin prime conjecture seems to be more permeable in this setting. (For instance, in my previous post I raised a possible approach, based on establishing expander properties of a certain random graph, which seems to get around the parity problem, in principle at least.)

(Update, Sep 25: fixed some treatment of error terms, following a suggestion of Andrew Granville.)

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